Abstract

In this paper, we suggest a hybrid method for finding a common element of the set of solution of a monotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method.

Keywords

1 Introduction

Let H be a real Hilbert space with inner product 〈· , ·〉 and induced norm || · ||. Let C be a nonempty closed convex subset of H. Let A : C → H be a nonlinear operator. It is well known that the variational inequality problem VI(C, A) is to find u∈C such that

⟨Au,v-u⟩≥0,∀v∈C.

The set of solutions of the variational inequality is denoted by Ω.

Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral and equilibrium problems, which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have been developed for solving variational inequalities and related optimization problems, see [1, 1–25] and the references therein. Let us start with Korpelevich's extragradient method which was introduced by Korpelevich [6] in 1976 and which generates a sequence {xn } via the recursion:

yn=PC[xn-λAxn],xn+1=PC[xn-λAyn],n≥0,

(1.1)

where PC is the metric projection from Rn onto C, A : C → H is a monotone operator and λ is a constant. Korpelevich [6] proved that the sequence {xn } converges strongly to a solution of V I(C, A). Note that the setting of the space is Euclid space Rn .

Korpelevich's extragradient method has extensively been studied in the literature for solving a more general problem that consists of finding a common point that lies in the solution set of a variational inequality and the set of fixed points of a nonexpansive mapping. This type of problem aries in various theoretical and modeling contexts, see e.g., [16–22, 26] and references therein. Especially, Nadezhkina and Takahashi [23] introduced the following iterative method which combines Korpelevich's extragradient method and a CQ method:

where PC is the metric projection from H onto C, A : C → H is a monotone k-Lipschitz-continuous mapping, S : C → C is a nonexpansive mapping, {λn } and {αn } are two real number sequences. They proved the strong convergence of the sequences {xn }, {yn } and {zn } to the same element in Fix(S) ∩ Ω. Ceng et al. [25] suggested a new iterative method as follows:

where A : C → H is a pseudomonotone, k-lipschitz-continuous and (w, s)-sequentially-continuous mapping, {Si}i=1N:C→C are N nonexpansive mappings. Under some mild conditions, they proved that the sequences {xn }, {yn } and {zn } converge weakly to the same element of ⋂i=1NFix(Si)∩Ω if and only if lim infn〈Axn , x - xn 〉 ≥ 0, ∀x∈C. Note that Ceng, Teboulle and Yao's method has only weak convergence. Very recently, Ceng, Hadjisavvas and Wong further introduced the following hybrid extragradient-like approximation method

for all n ≥ 0. It is shown that the sequences {xn }, {yn }, {zn } generated by the above hybrid extragradient-like approximation method are well defined and converge strongly to PF(S)∩Ω.

Motivated and inspired by the works of Nadezhkina and Takahashi [23], Ceng et al. [25], and Ceng et al. [27], in this paper we suggest a hybrid method for finding a common element of the set of solution of a monotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method.

2 Preliminaries

In this section, we will recall some basic notations and collect some conclusions that will be used in the next section.

Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping A : C → H is called monotone if

⟨Au-Av,u-v⟩≥0,∀u,v∈C.

Recall that a mapping S : C → C is said to be nonexpansive if

∥Sx-Sy∥≤∥x-y∥,∀x,y∈C.

Denote by Fix(S) the set of fixed points of S; that is, Fix(S) = {x∈C : Sx = x}.

It is well known that, for any u∈H, there exists a unique u0∈C such that

∥u-u0∥=inf{∥u-x∥:x∈C}.

We denote u0 by PC [u], where PC is called the metric projection of H onto C. The metric projection PC of H onto C has the following basic properties:

(i)

||PC [x] - PC [y] || ≤ ||x - y|| for all x, y∈H.

(ii)

〈x - PC [x], y - PC [x]〉 ≤ 0 for all x∈H, y∈C.

(iii)

The property (ii) is equivalent to

∥x-PC[x]∥2+∥y-PC[x]∥2≤∥x-y∥,∀x∈H,y∈C.

(iv)

In the context of the variational inequality problem, the characterization of the projection implies that

u∈Ω⇔u=PC[u-λAu],∀λ>0.

Recall that H satisfies the Opial's condition [28]; i.e., for any sequence {xn } with xn converges weakly to x, the inequality

liminfn→∞∥xn−x∥<liminfn→∞∥xn−y∥

holds for every y∈H with y ≠ x.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Si}i=1∞ be infinite family of nonexpansive mappings of C into itself and let {ξi}i=1∞ be real number sequences such that 0 ≤ ξi ≤ 1 for every i∈N. For any n∈N, define a mapping Wn of C into itself as follows:

So, ||xn- Wnxn || → 0 too. On the other hand, since {xn } is bounded, from Lemma 2.3, we have limn→∞||Wnxn- Wxn|| = 0. Therefore, we have

limn→∞∥xn-Wxn∥=0.

□

Finally, according to Conclusions 3.3-3.5, we prove the remainder of Theorem 3.1.

Proof. By Conclusions 3.3-3.5, we have proved that

limn→∞∥xn-Wxn∥=0.

Furthermore, since {xn } is bounded, it has a subsequence {xnj} which converges weakly to some ũ∈C; hence, we have limj→∞∥xnj-Wxnj∥=0. Note that, from Lemma 2.4, it follows that I - W is demiclosed at zero. Thus ũ∈Fix(W). Since tn=PCn[xn-λnAyn], for every x∈Cn we have

Since limn→∞(xn- tn) = limn→∞(yn- tn) = 0, A is Lipschitz continuous and λn≥ a > 0, we deduce that

⟨x-ũ,Ax⟩=limnj→∞⟨x-tnj,Ax⟩≥0.

This implies that ũ∈Ω. Consequently, ũ∈⋂n=1∞Fix(Sn)∩Ω That is, ωw(xn)⊂⋂n=1∞Fix(Sn)∩Ω.

In (3.8), if we take u=P⋂n=1∞Fix(Sn)∩Ω[x0], we get

∥x0-xn+1∥≤∥x0-P⋂n=1∞Fix(Sn)∩Ω[x0]∥.

(3.10)

Notice that ωw(xn)⊂⋂n=1∞Fix(Sn)∩Ω. Then, (3.10) and Lemma 2.5 ensure the strong convergence of {xn+1} to P⋂n=1∞Fix(Sn)∩Ω[x0]. Consequently, {yn } and {zn } also converge strongly to P⋂n=1∞Fix(Sn)∩Ω[x0]. This completes the proof.

Remark 3.5. Our algorithm (3.1) is simpler than the one in [23] and we extend the single mapping in [23] to an infinite family mappings. At the same time, the proofs are also simple.

Declarations

Acknowledgements

The authors are extremely grateful to the referees for their useful comments and suggestions which helped to improve this paper. Yonghong Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 100-2221-E-230-012. Jen-Chih Yao was partially supported by the Grant NSC 99-2115-M-037-002-MY3.

4 Competing interests

The authors declare that they have no competing interests.

5 Authors' contributions

All authors participated in the design of the study and performed the converegnce analysis. All authors read and approved the final manuscript.

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.